eBookISBN:  9781470423926 
Product Code:  CBMS/32.E 
List Price:  $26.00 
Individual Price:  $20.80 
eBook ISBN:  9781470423926 
Product Code:  CBMS/32.E 
List Price:  $26.00 
Individual Price:  $20.80 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 32; 1977; 41 ppMSC: Primary 11;
Knowledge about fractional parts of linear polynomials is fairly satisfactory. Knowledge about fractional parts of nonlinear polynomials is not so satisfactory. In these notes the author starts out with Heilbronn's Theorem on quadratic polynomials and branches out in three directions. In Sections 7–12 he deals with arbitrary polynomials with constant term zero. In Sections 13–19 he takes up simultaneous approximation of quadratic polynomials. In Sections 20–21 he discusses special quadratic polynomials in several variables.
There are many open questions: in fact, most of the results obtained in these notes ar almost certainly not best possible. Since the theory is not in its final form including the most general situation, i.e. simultaneous fractional parts of polynomials in several variables of arbitary degree. On the other hand, he has given all proofs in full detail and at a leisurely pace.
For the first half of this work, only the standard notions of an undergraduate number theory course are required. For the second half, some knowledge of the geometry of numbers is helpful. 
Table of Contents

Chapters

Heilbronn’s Theorem

The Heilbronn Alternative Lemma

Vinogradov’s Lemma

About Sums $\sum \ \xi _i \^{1}$

About Sums $\sum e(\alpha n^2)$

Proof of the Heilbronn Alternative Lemma

Fractional Parts of Polynomials

A General Alternative Lemma

Sums $\sum \ \xi _i \^{1}$ Again

Estimation of Weyl Sums

What Happens if the Weyl Sums are Large

Proof of the General Alternative Theorem

Simultaneous Approximation

A Reduction

A Vinogradov Lemma

Proof of the Alternative Lemma on Simultaneous Approximation

On max $\ \alpha _in^2 \$

A Determinant Argument

Proof of the Three Alternatives Lemma

Quadratic Polynomials in Several Variables

Proofs for Quadratic Polynomials


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Knowledge about fractional parts of linear polynomials is fairly satisfactory. Knowledge about fractional parts of nonlinear polynomials is not so satisfactory. In these notes the author starts out with Heilbronn's Theorem on quadratic polynomials and branches out in three directions. In Sections 7–12 he deals with arbitrary polynomials with constant term zero. In Sections 13–19 he takes up simultaneous approximation of quadratic polynomials. In Sections 20–21 he discusses special quadratic polynomials in several variables.
There are many open questions: in fact, most of the results obtained in these notes ar almost certainly not best possible. Since the theory is not in its final form including the most general situation, i.e. simultaneous fractional parts of polynomials in several variables of arbitary degree. On the other hand, he has given all proofs in full detail and at a leisurely pace.
For the first half of this work, only the standard notions of an undergraduate number theory course are required. For the second half, some knowledge of the geometry of numbers is helpful.

Chapters

Heilbronn’s Theorem

The Heilbronn Alternative Lemma

Vinogradov’s Lemma

About Sums $\sum \ \xi _i \^{1}$

About Sums $\sum e(\alpha n^2)$

Proof of the Heilbronn Alternative Lemma

Fractional Parts of Polynomials

A General Alternative Lemma

Sums $\sum \ \xi _i \^{1}$ Again

Estimation of Weyl Sums

What Happens if the Weyl Sums are Large

Proof of the General Alternative Theorem

Simultaneous Approximation

A Reduction

A Vinogradov Lemma

Proof of the Alternative Lemma on Simultaneous Approximation

On max $\ \alpha _in^2 \$

A Determinant Argument

Proof of the Three Alternatives Lemma

Quadratic Polynomials in Several Variables

Proofs for Quadratic Polynomials